Inclusion problems for patterns with a bounded number of variables

We study the inclusion problems for pattern languages that are generated by patterns with a bounded number of variables. This continues the work by Freydenberger and Reidenbach [D.D. Freydenberger, D. Reidenbach, Bad news on decision problems for patterns, Information and Computation 208 (1) (2010) 83–96] by showing that restricting the inclusion problem to significantly more restricted classes of patterns preserves undecidability, at least for comparatively large bounds. For smaller bounds, we prove the existence of classes of patterns with complicated inclusion relations, and an open inclusion problem, that are related to the Collatz Conjecture. In addition to this, we give the first proof of the undecidability of the inclusion problem for NE-pattern languages that, in contrast to previous proofs, does not rely on the inclusion problem for E-pattern languages, and proves the undecidability of the inclusion problem for NE-pattern languages over binary and ternary alphabets.